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Question Number 181707 by amin96 last updated on 28/Nov/22

$$\mathit{i}\mathit{f}2\mathit{x}+\frac{1}{\sqrt{\mathit{x}}}=\frac{1}{2}$$$$\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{t}\mathit{h}\mathit{e}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}8\mathit{x}+\frac{1}{\sqrt{\mathit{x}}}=?$$

Answered by mr W last updated on 29/Nov/22

$$t=\sqrt{x}>0$$$$f\left(t\right)=2t^2+\frac{1}{t}$$$$\frac{df}{dt}=4t-\frac{1}{t^2}=0\Rightarrow t=\frac{1}{\sqrt[3]{4}}$$$$f_{min}=\frac{1}{\sqrt[3]{2}}+\sqrt[3]{4}\approx 2.381>\frac{1}{2}$$$$\Rightarrow 2x+\frac{1}{\sqrt{x}}=\frac{1}{2}isnotpossibleforx\in R.$$

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